Mathematical Pattern Poetrym.archive.bridgesmathart.org/2012/bridges2012-65.pdf · results in...
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Mathematical Pattern Poetry
Sarah Glaz
Department of Mathematics
University of Connecticut
Storrs, CT 06269, USA
E-mail: [email protected]
Abstract
Pattern poetry is an art form that combines literary and visual elements to produce poems in which the text and the
shape of the poem work together as an aesthetic whole. To fully appreciate a pattern poem one needs to see it as
well as hear it. Although varied forms of this art existed throughout history and across cultures, it is only in the 19th
century that pattern poetry became a recognized genre of poetry. This genre is further enriched by the increasing
graphing capabilities of modern computers that add a new dimension to the hand-drawn or typewriter-produced
pattern poems of the past. Mathematical pattern poetry―pattern poetry possessing a substantial mathematical
component―includes poems whose shape or content involve geometric figures, mathematical curves, or other
mathematical notions and symbols. This article provides a brief overview of modern mathematical pattern poetry,
including selected references for the use of such poetry in the mathematics classroom. The concluding remarks
touch lightly on the poetry forms that sprang from this genre: visual poetry and electronic poetry.
Plane Figures and Curves
One of the earliest known pattern poems, The Sator Square [28], has a
mathematical shape. It is a 5x5 letter-square containing the Latin palindrome
SATOR AREPO TENET OPERA ROTAS. Representations of The Sator Square
were found in numerous European archeological sites, the earliest going back to
the ruins of Pompeii, which was buried under volcanic ash in 79 AD. A rough
translation is: “The farmer Arepo uses his plough as his form of work.” The
significance of the square is mysterious. Speculations connect it to various letter
rearrangements which lend it an early Christian religious association. Later square
poems were constructed as labyrinths or involved other internal puzzle-patterns
that were religious in nature [15]. An interesting example is the 16th century 10x10 syllable-square from
Sundry Christian Passions by Henry Lok, Square Poem in Honor of Elizabeth I [11, 34].
Below is a modern square poem by JoAnne Growney, which appears in [12] for Earth Day, April 21,
2011. It is an 8x8 syllable-square poem concerned with the depletion of one of our most important
resources―water. Mathematical in both shape and content, the
poem describes an exponential-rate consumption of water. But
one does not need to understand the mathematics of
exponential growth in order to fully absorb the message of this
poem. Substituting “syllable” for “quart” and doing the
calculation described in the poem, one can practically see the
water level dropping as the poem disappears syllable by
syllable: On the first day 1 syllable is gulped; on the second
day 2 syllables are gulped, making the total number of gulped
syllables after two days 1+2=3; the total number of gulped
syllables after 3 days is 1+2+22=7. How long before no water is
left? Gone in a week…. In addition to aesthetic value this poem
is particularly suited for use in an algebra class to enhance the
Square Math Problem
by JoAnne Growney
Quietly the dark creature starts--
it drinks a quart of the water
from our reservoir. Then each day
it gulps twice as much as the day
before. If no one notices
this monster’s thirst until one-fourth
the water’s gone, what time is left
to arrest the vast consumption?
Bridges 2012: Mathematics, Music, Art, Architecture, Culture
65
teaching of the topic of exponential growth and decay. Poetry projects are used in the mathematics
classroom for their power to engage attention and enhance memory. Careful project construction often
results in additional pedagogical benefits, such as better integration of material and easier transition to its
applications. Ideas on how to construct poetry projects and a survey on the use of poetry in college
mathematics classes may be found in [7, 8, 9,10].
To the right is a pattern poem by Li C. Tien, Right Triangle Right Triangle
[33]. It involves squares and triangles, although these shapes are by Li C. Tien
not made up by counting syllables. The image brings to mind the
various proofs of Pythagoras Theorem [2] and, like the previous
poem, Right Triangle could be used to enhance the teaching of
this topic in an algebra class. Squares and triangles are not the
only two dimensional figures used to construct pattern poems.
Gerald Lynton Kaufman’s Geometric Verse: Poetry Forms in
Mathematics Written Mostly for Fanatics [18] contains pattern
poems involving rectangles, rhombuses, trapezoids, polygons,
circles, ellipses, and much more. Other sources of pattern poetry
involving two dimensional figures are [1, 4, 11, 14, 15, 27].
We conclude our discussion of pattern poems whose shape
involves planar figures with two poems involving curves. Below,
on the right, is: Riddle in the Shape of an Octagon, by the
Mexican Nobel laureate poet Octavio Paz [24]. It involves 8 line
segments radiating from a center―an image that can be viewed as
a division of an octagon. Following the poet’s instructions one can
reorder and place the 8 statements appearing under the geometric
figure in the octagon’s “slices” to read an enchanting poem. Try
it! Lines also appear in Eugene Guillevic’s poems: Line, Parallels,
and Perpendicular [14]. On the left is Gerald Lynton Kaufman’s
Spiralay [18], whose shape is a spiral. Spirals are graphs of certain functions in polar coordinates, such as,
for example, the logarithmic spiral [1]. Spirals as shapes for poems also appear in The
Derivative, by Betsy Adams [26], and Spiral by Eugene Guillevic [14]. The latter also includes other
curve patterns such as parabolas, cycloids, sinusoids, and hyperbolas.
Spiralay Riddle in the Shape of an Octagon
by Gerald Lynton Kaufman by Octavio Paz
Glaz
66
Solids, Möbius Strips, and Fractals
In a cylindrical pattern poem called Iambi-Cylinder, Gerald Lynton Kaufman [18] writes:
iambic cylinders are seldom found
because their music goes around and round
so publishers who read them never know
what’s printed on the side that does not show
In other words, poets who try to write a pattern poem with a three dimensional shape, or a two
dimensional shape that does not lie flat on the plane, encounter the same difficulties mathematicians have
when trying to draw three dimensional figures on a flat piece of paper. It is interesting to note that there
is more than one solution this problem.
The lovely little poem to the left, The Truncated Cone by
Truncated Cone Eugene Guillevic [14], provides one of the simplest solutions―
by Eugene Guillevic place the poem and the figure representing its shape separately―
but in close proximity. Effective, but perhaps not satisfying the
impulse to integrate the text and the form of the poem.
A different solution is offered in Tyehimba Jess’ poem, The
Bert William/George Walker Paradox [17]. The poem is displayed
on the page in two equal size columns: the left column for Bert
Williams’ words, and the right column for George Walker’s words.
The reader is advised to cut the poem’s text out of the page and
roll, fold, and glue as necessary to read the poem on cylinders or on
a Möbius strip. We reproduce a portion of the poem and
summarize the reading instructions below. As background to the
poem the poet provides the following introduction: “Bert Williams
and George Walker were an African American comedy duo who,
through extraordinary skill and insight, were able to depict
subtlety, humanity and intelligence within the confines of the only
site black comedic actors could work at the turn of the 19th to 20
th
century―the minstrel show. Bert Williams (1874-1922) became
known as “The Jonah Man” and was known for his trademark song “Nobody.” George Walker (1873-
1911) was known for his dancing skills and played the straight man in their act.”
From: The Bert William/George Walker Paradox
by Tyehimba Jess
………………………………………………………………………………………………………………
Sing like me, Jonah in the charcoal hold of the whale―
be smiling till the crowd starts to cry: this nobody
born to riddle all ‘bout race. A sure bet nobody
might be saying “look at that handsome man!” Nobody
wondering bout dark’s pathos. Believe the human
being buried ‘neath my ash. Don’t claim that nobody
holding secret smiles for another man’s plight. A shame
knuckled under humor. Who escapes it? Nobody
Doing justice to my Juba jig. See how I dance?
just might be playing you for a fool. You see this face
got them laughing loud like me. You think they mock my face?
Can’t help but see themselves beneath my blacked up face―
being swept up in my diamond stud strut: Bless those that
getting touched from inside out. Every wide eyed face
in the crowd all gut-bent with laughter―they know that they
will be baptized clean. I want to make it so―each face
………………………………………………………………………………………………………………
Mathematical Pattern Poetry
67
Below is a summary of the instructions given by the poet for turning the poem into a pattern poem:
1. Cut the poem’s text along the four dotted lines.
2. To read the poem on a horizontal cylinder roll the poem’s text and glue the top
and bottom dotted lines.
3. To read the poem on a vertical cylinder roll the poem’s text and glue the two
side dotted lines.
4. To read the poem on a band, fold the poem’s text along the line separating the
columns. Roll the resulting half text and glue top to bottom. You may now read the
poem on the circular band from the inside as well as the outside.
5. To read the poem on a Möbius strip, after folding the
poem’s text down the middle as in step 4, give it a half twist
before you glue the ends. The poem can now be read along
a two dimensional surface with only one side―a Möbius
strip. To the right is the poem placed on a Möbius strip.
Another solution for presenting a three dimensional pattern
poem on the flat page is to provide an illusion of three
dimensionality. This can be achieved through various techniques.
Gerald Lynton Kaufman [18], who was an architect, achieved this
effect through the use of perspective when drawing the shape and the letters of the poems in a way similar
to its use for drawing architectural diagrams. A particularly ingenious example is his poem Cubicouplets
[18], which we cannot include here because of space limitations. The poem manages to present the words
written on both the seen and the “unseen” side of a cube.
My poem to the left, Do You Believe in Fairy Tales?, employs
Do You Believe in Fairy Tales? a different technique to create an illusion of a three dimensional
by Sarah Glaz spherical object. The object―a planet or a moon―is evoked through
typography. The poem’s lines are arranged in a way that their tips
draw a parabola and the resulting image displays characteristics that
are universally associated with moon images. The reader’s
imagination does the rest. Caleb Emmons poem, Seeing Pine Trees
[6], is another mathematical pattern poem employing such technique
for the visualization of a non-mathematical three dimensional
object―an evergreen.
We conclude this section with another pattern poem that employs
illusion for its presentation. The poem, The Cantor Dust ― a Fractal
Poem, by Rodrigo Siqueira [30], which appears on the next page,
uses the visual structure of the Cantor Set to send a poetic and
mathematical message. The Cantor Set is constructed by repeatedly
deleting the middle third from each interval of a growing collection of
intervals. The construction starts with the deletion of the middle third
from the interval [0, 1]. Then, the middle third from each of the two
remaining intervals is deleted. Next follows the deletion of the middle
third from each of the four remaining intervals, and so on. The
process never terminates. Although this process does not yield a three
dimensional object it is difficult to depict it in any dimension because
it never ends. Siqueira’s poem uses shape, size, shading, empty spaces
and the reader’s imagination to fill in the gap between the finite and
the infinite. The sources cited in this section and [26] contain many
other interesting examples of three dimensional mathematical pattern
poems.
Glaz
68
The Cantor Dust ― a Fractal Poem
by Rodrigo Siqueira
Beyond Geometry and Topology
In addition to geometric and topological patterns, mathematics is rich in symbols, notation, and various
possibilities of manipulating entities that can be used as patterns for poems. Given the place numbers hold
in mathematics, it is perhaps surprising that there is no abundance of poems utilizing the shapes of
numbers. Still, such poems exist. Below is a small fragment from the book length poem Midpoint by John
Updike [35], where numbers paint a more vivid image from one’s childhood than any number of words
can do. A poem shaped like number 1 by Thomas
From: Midpoint: The Play of Memory Eisele appears in [5]. Other poems utilizing numbers
by John Updike appear in [26].
The symbol signifying the “end” of the number
line, the horizontal eight―infinity―appears either on
its own or in correlation with other numbers in
several poems of [26] and [18]. Below is Bernhard
Frank’s Infinity [26], a poem that comments on its
own shape as well as on a certain difficult human
condition.
Infinity by Bernhard Frank
Mathematical Pattern Poetry
69
The last group of mathematical pattern poems we will discuss in this article involves symbols
and words that denote mathematical objects or operations, in a variety of surprising and delightful
ways. Below, to the left, is a small fragment from Lois Zukofsky A series poems [36]. It employs the
mathematical symbol for an integral to give a concise explanation of his poetics as the integration of
speech and music. The integral sign explains visually what the words below it, using the
mathematical language of upper and lower limits of integration, do verbally. The combination is
aesthetically pleasing in the way a bilingual version of an original poem and its good translation is
pleasing for a reader fluent in both languages. Similarly, Bob Grumman’s Mathemaku No. 10 [11,
13] employs mathematical notation to extend the capability of language and express the poem’s
intent in a more compact, minimalist, way. Grumman went one step further than Zukofsky and
omitted some of the words needed to fully explain the mathematical iconography. This adds an
intriguing layer of ambiguity, but in the same time sacrifices the opportunity of making music
through speech. Mathemaku No. 10 is one of the early mathemakus, mathematical haikus, created by
Bob Grumman. His mathemakus evolved over time to become more visual by including more
symbols and images and less words. This trend from pattern poetry to what is called nowadays visual
poetry, is apparent in the works of other poets who create visual poetry―a subject we will return to
in the concluding remarks. The third poem in this grouping, using an even smaller amount of words,
is Kaz Maslanka’s, Sacrifice and Bliss [11, 21]. In addition to mathematical notation this poem
involves a mathematical notion which requires the reader to actually “do mathematics” on the words
in order to understand the poem. Specifically, as the denominator of the fraction, the ego, shrinks to
0, the fraction grows larger and larger, till―at the limit―it becomes infinite. In other words, love’s
strength is inversely proportional to the size of the ego. Maslanka’s poetry [21], which he names
equation poetry, involves several other mathematical operations. He also embeds many of his poems
in computer enhanced visual environments that intensify their visual impact.
From: A 12 Mathemaku No. 10
by Louis Zukofsky by Bob Grumman
Sacrifice and Bliss The last poem in this group, 26 Points to Specify [25], by
by Kaz Maslanka the French surrealist poet Benjamin Péret, appearing on the
next page, uses mathematical symbols to extend the capability
of language in a different way from the three poems mentioned
above. The mathematical formulas at the end of each of the
poem’s 26 lines complement what is said by what is
unsayable―or by that which is better left unsaid. If I may
venture an interpretation, the ever increasing complexity of the
formulas as the poem progresses backwards from death to birth
Glaz
70
is meant to express the richness of the gifts
with which we are born and their gradual
diminishment as we age.
Several other poets employ
mathematical symbols and formulas as a
partial pattern embedded in a traditional
poem in order to extend the capability of
language. The reasons for using this kind of
pattern and its meaning in any particular
poem are complex, and vary from poem to
poem. Examples of poems using this
technique include: Ray Bobo’s Give Me an
Epsilon and I will Treat It Well [11], Lewis
Carroll’s Yet What Are All Such Gaieties
[3], Velimir Khlebnikov’s Order from the
Presidents of Planet Earth [19], H. K.
Norla, Of Xs and of Ys [23], Ed Seikota’s
Borderline―a Fractal Poem [11, 29], C.K.
Stead’s Walking Westward [11], Stephanie
Strickland’s Gibbous Statement [32], and
my poem, The Enigmatic Number e [8].
Concluding Remarks
Pattern poetry is a well-established literary term with a significant history. In addition to Lok’s 16th
century square poem mentioned in the first section, there are other classical pattern poems authored by,
for example, Simmias of Rhodes (c. 325 BC) and George Herbert (1593–1633) [15, 34], possessing
hidden mathematical dimensions. More recent, the 1897 poem: A Dice Throw by Stéphane Mallarmé
[20], does as well. Due to space restrictions, we could not include an exploration of these intriguing
mathematical aspects. Neither could we discuss the multitude of mathematical patterns generated by
OULIPO (Ouvroir de Littérature Potentielle)―the literary movement founded in France by Raymond
Queneau in 1960 [22]. OULIPO’s patterns, among them the Möbius strip pattern of Jess’s poem
appearing in the second section, are used to construct literature to this day. The division of mathematical
pattern poems into the three groups corresponding to this article’s three sections, is a first step towards a
classification of this poetry. Further refinements would have to take these omissions into account.
Pattern poetry is also called shape poetry, concrete poetry, or visual poetry. As the small sample
presented in this article shows mathematical pattern poetry covers a wide range of possibilities, not all
fitting comfortably under the umbrella of any of the names. The name “shape poetry” does not fit well
poems whose shape is not geometric or topological, and the name “concrete poetry” does not fit well
poems that involve the more abstract―the very non-concrete―symbols of mathematics. “Visual poetry”
is a relatively new term which is better reserved for a form of poetry that evolved from pattern poetry.
The term “visual poetry” describes best poetry which places more emphasis on the visual aspect of the
poems than on their word-induced musicality. Good sources to learn more about this form of poetry are
Geof Huth’s article [16] and Kaz Maslanka’s blog [21]. I picked the term “pattern poetry,” as it seems to
describe best the kind of poems selected for this article―poems for which literary musicality is as
important as visual pattern.
In addition, it is problematic to call a poem “a mathematical poem.” Maslanka’s blog [21] presents
a friendly, but heated, debate on what exactly constitutes mathematical poetry. To read the debate search
the site using the words “Types of Mathematical Poetry.” My choice is to be inclusive and call a poem
mathematical if it has a significant mathematical component of any kind.
From: 26 Points to Specify
by Benjamin Péret
My life will end by
I am
I ask for
I consider the religious holidays
My prediction for the future
.
.
.
My fortune (
( √
( )
)
√( ) )
The date of my birth (
( √
( )
)
√( ) )
Mathematical Pattern Poetry
71
Visual poetry is not the only poetry form that evolved from pattern poetry. With the increasing
capabilities of computers, it was only a matter of time, before artists introduced animation, interactivity,
and even sound into their work. Visual or traditional poetry with interactive, animated, or sound elements
is computer code driven. This kind of poetry is called electronic poetry. A good source to learn more
about electronic poetry is an article by Stephanie Strickland [31].
Acknowledgements
The author gratefully acknowledges permissions to reprint from: J. Growney for "Square Math Poem;” L.C. Tien for "Right
Triangle;” R.Sieburth, translator, for “Truncated Cone” by E. Guillevic; B.Grumman for “Mathemaku No. 10;” K. Maslanka for
“Sacrifice and Bliss;” T. Jess for fragment from “The Bert William/George Walker Paradox;” B. Frank for “Infinity;” New
Direction Publishing Corp. for “Riddle in the Shape of an Octagon” by O. Paz, translated by E. Weinberger; R. Siqueira for “The
Cantor Dust―a Fractal Poem;” Oberlin College Press for fragment from “26 Points to Specify” by B. Péret, translated by K.
Hollaman, copyright ©1981 by Oberlin College.
References [1] M. Birken & A.C. Coon, Discovering Patterns in Mathematics and Poetry, Editions Rodopi, 2008
[2] A. Bogomolny, Pythagoras Theorem, http://www.cut-the-knot.org/pythagoras/index.shtml
[3] L. Carroll, The Complete Works of Lewis Carroll, Penguin Books, 1988
[4] J. Earls, The Lowbrow Experimental Mathematician, Pleroma Publications, 2010
[5] T. Eisele, Untitled, Poetry East 42, p 43, Spring 1996
[6] C. Emmons, Seeing Pine Trees, Journal of Humanistic Mathematics 1, 2011
[7] S. Glaz & S. Liang, Modeling with Poetry in an Introductory College Algebra Course and Beyond, Journal of Mathematics
and the Arts 3, p 123-133, 2009
[8] S. Glaz, The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom, MAA Loci: Convergence,
DOI 10.4169/loci 003482, 2010
[9] S. Glaz, The Poetry of Prime Numbers, Proceedings of Bridges 2011, p 17-24, 2011
[10] S. Glaz, Poetry Inspired by Mathematics: A Brief Journey through History, Journal of Mathematics and the Arts 5, p 171-
183, 2011
[11] S. Glaz & J. Growney (ed.), Strange Attractors: Poems of Love and Mathematics, CRC Press/A K Peters, 2008
[12] J. Growney, Intersections – Poetry with Mathematics, http://poetrywithmathematics.blogspot.com
[13] B. Grumman, Mathemaku 6-12, http://www.thing.net/~grist/l&d/grumman/lgrumn-1.htm
[14] E. Guillevic, Geometries, R. Sieburth, tr., Ugly Duckling Presse, 2010
[15] D. Higgins, Pattern Poetry, State University of New York Press, Albany, 1987
[16] G. Huth, Visual Poetry Today, Poetry, November 2008. Poetry Foundation:
http://www.poetryfoundation.org/poetrymagazine/article/182397
[17] T. Jess, The Bert Williams/George Walker Paradox, The American Poetry Review 40, p 44-45, 2011
[18] G. L. Kaufman, Geo-Metric Verse: Poetry Forms in Mathematics Written Mostly for Fanatics, The Beechhurst Press, 1948
[19] V. Khlebnikov, Collected Works of Velimir Khlebnikov, Vol. III, P. Schmidt, tr., Harvard University Press, 1997
[20] S. Mallarmé, Collected Poems and Other Verse, E.H. Blackmore & A.M. Blackmore, tr., Oxford University Press, 2009
[21] K. Maslanka, Mathematical Poetry Blog Spot, http://mathematicalpoetry.blogspot.com/
[22] H. Mathews & A. Brotchie, Oulipo Compendium, Atlas Press, 1998
[23] H. K. Norla, Of Xs and of Ys, http://hknorla.blogspot.com/2005/09/today-mathematical-poet.html
[24] O. Paz, The Collected Poems of Octavio Paz, 1957-1987, E. Weinberger, tr., New Directions, 1987
[25] B. Péret, From the Hidden Storehouse: Selected Poems of Benjamin Péret, K. Hollaman, tr., Field Translation Series 6,
Oberlin College Press, 1981
[26] E. Robson & J. Wimp (ed.), Against Infinity, Primary Press, 1979
[27] A. Saroyan, Complete Minimal Poems of Aram Saroyan, Ugly Duckling Presse, 2007
[28] Sator Square, http://en.wikipedia.org/wiki/Sator_Square
[29] E. Seykota, Borderline―a Fractal Poem, http://www.seykota.com/tribe/fractals/index.htm
[30] R. Siqueira, The Cantor Dust―a Fractal Poem, http://www.insite.com.br/rodrigo/misc/fractal/fractal_poetry.html
31] S. Strickland, Born Digital, Poetry Foundation: http://www.poetryfoundation.org/article/182942
[32] S. Strickland, Zone: Zero, Ahsahta Press, Boise State University, 2008
[33] L. C. Tien, Right Triangle, The Mathematical Intelligencer 31, p 50, 2009
[34] UBU Web: Historical, http://www.ubu.com/historical/
[35] J. Updike, Collected Poems of John Updike, 1953-1993, Alfred A. Knopf, 1993
[36] L. Zukofsky, “A” 1-12, Doubleday & Co., 1967
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